TowhomcorrespondenceshouldbeaddressedEmailbarneybalesbarneybalesaq ID: 121857
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PrecisionRelativeAggregationNumberDeterminationsofSDSMicellesUsingaSpinProbe.AModelofMicelleSurfaceHydrationBarneyL.Bales,*LuisMessina,ArwenVidal,andMiroslavPericDepartmentofPhysicsandAstronomyandTheCenterforCancerandDeelopmentalBiology,CaliforniaStateUniersityatNorthridge,Northridge,California91330-8268OtaciroRangelNascimentoDepartamentodeFõsicaandInformatica,InstitutodeFõsicadeSaoCarlos,UniersidadedeSaoPaulo,CEP13560-970SaoCarlos,Braziled:August13,1998Aspin-probemethodisdescribedthatcandetectchangesintherelativeaggregationnumbersinSDSmicelleswithaprecisionofaboutonemolecule.ThemethodisbasedonthefactthattheNhyperfinecouplingconstantissensitivetotheaveragefractionofthevolumeoccupiedbywaterintheregionofthenitroxidemoietythatislocatedonaveragenearthemicellesurface.Defining)tobetheNhyperfinecouplingconstantatanaggregationnumber,wefind,whereiscontrolledbyvaryingeithertheSDSortheNaClconcentrations.Forthespinprobe5-doxylstearicacidester(5DSE),bycombinationoftheresultsofexperimentsinwhichtheSDSand/ortheNaClconcentrationswerevaried,linearleast-squaresfitsgave0.009)Gand0.02mG/molecule)dependsonlyontheaggregationnumberdespitethefactthatagivenvalueofmaybepreparedbychoosingdifferentcombinationsofNaClandSDSconcentrations.Thismeansthat,foragivenmicellesize,neitherinteractionsbetweenmicellesnortheionicstrengthofthesolutioninfluencethevalue.Ageometricmodel,basedonasimplemodelinwhichasphericalhydrocarboncoreissurroundedbyaconcentricsphericalpolarshell,isdevelopedtopredictthevolumefractionofthepolarshelloccupiedbywaterasafunctionof.Thiswaterfractioniswrittenintermsofahydrationnumberpersurfactant)isrelatedtoO)byemployingthenonempiricalpolarityindexintroducedbyMukerjeeetal.(Mukerjee,P.;Ramachandran,C.;Pyter,R.A.J.Phys.Chem,3189).ThevalueO)decreasesasthemicellegrowsbecausethevolumepersurfactantmoleculeinthepolarshelldecreases.OnceO)isspecifiedataparticularvalueof,nofurtheradjustableparametersenterintothemodel.ThevariationwithmicellesizeofthetheoreticalpolarityisinexcellentagreementwithexperimentforvaluesofO)thatarecomparabletothosefoundfromtransportproperties.Thesphere-rodtransitionisobservedasarathersharptransitioninwhichthedetectedwatervolumefractionbecomesconstant.Detailedfuturetestsofthemodelareoutlined.Whenpreparedfrompuresurfactantsinpurewaternearthecriticalmicelleconcentration(cmc)(seeTable1fornomen-clatureandabbreviations),mostionicmicelleshavewell-definedaggregationnumbers,;i.e.,thesizedispersionissmall.InthecaseofSDS,increasingthesurfactantconcentrationaddingsaltoravarietyofotherimpurities(see,forexampleref5andreferencestherein)causesthemicellestogrowtolargeraggregationnumbers,Fortheseriesofnormalsodiumalkylsulfatesfromoctyltododecyl,Huismanfoundanempiricalrelationshipbetweenandthecmcthatwasindependentofboththealkylchainlengthandtheconcentrationofaddedcommoncounterion.Incorpo-ratingresultsduetoSasakietal.,Quinaetal.madethesuggestionthatthisempiricalrelationshipcouldbeextendedtosurfactantconcentrationsabovethecmcasfollows:where[NaisthemolarconcentrationofsodiumionintheaqueousphasewhetheritisprovidedbySDSorbothSDSandaddedsaltandareconstantsthatdependonthealkylchainlength.ForSDS,Quinaetal.wentontoshowthatexperimentalresultsfromawidevarietyoftechniquesfitanequationoftheformineq1quitewell;however,therewasaspreadinthevaluesofThecontributionsto[Nafromthesurfactantandaddedsalt,NaClinthiswork,maybefoundfromtheconventionalpseudophaseionexchangemassbalancerelationshipwherethebracketsindicatemolarconcentrations,istheapparentdegreeofcounteriondissociation,,andandfreeistheconcentrationofmonomericSDS.Themolar *Towhomcorrespondenceshouldbeaddressed.E-mail:barney.bales@barney.bales@+]aq)ç(1)[Na+]aq)R([SDS]-[SDS]free)+[SDS]free+[Na+]ad)R[SDS]+â[SDS]free+[NaCl](2)J.Phys.Chem.B10.1021/jp983364aCCC:$15.001998AmericanChemicalSocietyPublishedonWeb11/14/1998 concentrationofaddedsodiumion,[Na,isequalto[NaCl]inthiswork.Combiningeqs1and2yields,wemayrewriteeq3asfollows:When[SDS][SDS]freeand[NaCl]0,thefractioninthebracketsbecomesunityandisseentobetheaggregationnumberatthecmcintheabsenceofaddedNaCl.Weadopt0.27,whichisbasedonreliableactivityToconformtothevaluespredictedbyQuinaetal.,wealsoadoptthevalues164andasderivedfromHuisman'sempiricalequation.Usingcmc0.0083Mresultsin49.5.Thus,eq4becomesInwritingeq5,wehavetacitlyassumedthatisconstant,whichhasonlybeendemonstrateduptovaluesof[SDS]mM.Smallvariationsinwouldnotbeeasytodetectindirectmeasurementsofgiventheuncertaintiesinthevariousmethods,especiallyatlargeSDSconcentrations.Weemployeq5asaconvenientmethodtopresenttheresultsandshowbelowthattheconclusionsofthisworkarethesamewhetherweuseeq5ordirectlymeasuredvaluesof.Valuesofoffreearecalculatediterativelyfromeq5ofref8.Themotivationforthepresentworkarosefromobserva-thattheisotropichyperfinecouplingconstant,,ofthenitroxidespinprobe5-doxylstearicacidester(5DSE)variedlinearlywith[SDS].Bezzobotnovetal.,onthebasisofsmall-angleneutronscattering(SANS)experiments,hadsug-gestedthatvariedlinearlywith[SDS],leadingustothatmeasurementsofmightleadtoaneasy,precisemethodtoinvestigate.Notethattheproposedproposed1/4dependenceofintheabsenceofNaClisidenticaltoeq4butexcludingtheterminvolving[SDS].Thedifferenceinthedependenceofvs[SDS]includingorexcluding[SDS]onlybecomesapparentatsurfactantcon-centrationsapproachingcmc.Thederivationofeq3aspresentedinref8,experimentaldata,andtheresultsgiveninthisworkallfavortheinclusionof[SDS]Itiswell-knownthatforanitroxidespinprobesuchas5DSEvariessystematicallywiththesolventpolarity,thepreviousworkshowedthatasthemicellegrewbecauseofincreasing[SDS],thepolaritysensedby5DSEdecreased.Thefirstpurposeofthepresentworkistoshowthatthisbehavioristruewhetherthesizeisvariedbychangingthesurfactantorthesaltconcentrations;i.e.,isthesameforagivenvalueofevenifthisaggregationnumberisattainedusingdifferentcombinationsof[SDS]and[NaCl].Thispurposeseemsworthpursuingfortworeasons.(1)isasimplequantitytomeasurewithhighprecision.Withmodernmagnetpowersuppliesandcontrollersprovidingreproducible,linearmagneticfieldsweepsandwithspectralfittingtechniques,bemeasuredwitharelativeprecisionofatleast2orders-of- TABLE1:NomenclatureandAbbreviationsSDSsodiumdodecylsulfate5DSEnitroxidespinprobe5-doxylstearicacidestercmccriticalmicelleconcentrationcmcintheabsenceofaddedsaltorotheradditivesnumber-averagedaggregationnumberofSDSmicellesnumber-averagedaggregationnumberofSDSmicellesatthecmcapparentdegreeofcounteriondissociation;)one-halfthedifferenceintheresonancefieldsofthehigh-andlow-fieldresonances,neglectingsecond-orderdynamicshifts,equaltotheNhyperfinecouplingconstant.Alsowrittendifferenceintheresonancefieldsofthelow-andcenter-fieldresonancesdifferenceintheresonancefieldsofthehigh-andcenter-fieldresonancessecond-ordershiftparametersecond-orderdynamicshiftsin,respectivelyrotationalcorrelationtimeofthenitroxidemoietyC)polarityindexat25C;ratioofthemolarconcentrationofOHdipolestothatinwater;forSDSmicelles,thenumberdensityofwatermoleculesnormalizedtothatinpurewaterwater+]aqmolarconcentrationofsodiumionintheaqueouspseudophasepseudophase+]admolarconcentrationofaddedsodiumion;[Na[NaCl]inthispaper.paper.freemolarconcentrationofmonomericSDSprojectionofthenuclearspinofNlabelingthelines;1,0,and1forthelow-,center-,andfieldlines,respectivelyresonancefieldsofthethreelines1,0,andvolume(…)occupiedbyonesaturatedhydrocarbonchainnumberofcarbonatomsperchaininthehydrocarboncorevolume(…)persurfactantmoleculeinthepolarshellvolume(…)persurfactantmoleculeinthepolarshellinaccessibletowaterradius(…)ofhydrocarboncoreradius(…)ofmicelle0)numberofwatermoleculespersurfactantmoleculeinthepolarshell;thehydrationnumberSANSsmall-angleneutronscatteringTRFQtime-resolvedfluorescencequenchingweightfractionofmethanolinwatermethanolmixturesmolarratioof5DSEtoSDSmolarratioof5DSEtomicellizedSDS cmc0}ç(4)NA)49.5{0.27[SDS]+0.73[SDS]free+[NaCl] 0.0083}0.25(5) J.Phys.Chem.B,Vol.102,No.50,1998Balesetal. magnitudebetterthanmeasurementsofitself.Thus,itpotentiallyoffersasimplemethodtodetectsmallchangesin.(2)Thetheoreticalbasisforvariationsiniswelltherefore,themethodcouldleadtoamodelthatprovidesbetterinsightintothelocalstructureofthesolubilizationsiteofthespinprobe.ThesecondpurposeoftheworkistoproposesuchamodelforthesurfacehydrationoftheSDSmicelle.ThesuitablilityoftheuseofthesolvatochromicpropertiesofnitroxidefreeradicalsforstudyinglipidassembliessuchasmicelleshasbeencriticallydiscussedbyMukerjeeandco-workers.Seerefs13,1621,23,and24andreferencesthereinforthoroughdiscussionsofthemechanismsleadingtothesolvatochromicproperties.Mukerjeeetal.introducedanonempiricalpolarityscalethatshowedexcellentlinearcorrelationwithvaluesoffortwonitroxidespinprobes.Thescale,referredtoasapolarityindexanddenotedbyC)bythoseauthors,isdefinedtobetheratioofthemolarconcentrationofOHdipolesinasolventorsolventmixturetothatinwater.C)showsgoodlinearcorrelationwithothermeasuresofpolarity,asthedielectricconstant,withtheobviousadvantagethatitisastraightforwardmattertocalculatetheindexfrommolecularstructuresanddensities.ProvidedthatthesolventmixtureofinterestiswaterandmoleculespossessingnoOHbonds,C)issimplythevolumefractionoccupiedbywater.ThegrowthoftheSDSmicellesisaccompaniedbyanincreaseintheirsizedispersion,which,inprinciple,requiresadistinctionbetweennumber-ormass-averagedaggregationnumbers.Thetwoaveragesarerelatedasfollows:,wherethesubscriptsdenotemass-averagedandnumber-averagedaggregationnumbers,respectively,andisthedispersioninthenumber.Inpractice,thisdispersionreachesabout22moleculesfor[SDS]70mMand[NaCl]400mM,leadingtoamaximumdifferenceinthetwoaveragesoflessthanfourmolecules.Allofourmeasurementsfromwhichwederivedquantitativeresultsliewithinthisrange.Sincetheuncertaintiesinareontheorderoffivemolecules(seebelow),weignorethedifferenceinthetwoaverages.TheaggregationnumbersofSDSmicellesarealsotemperature-CombiningresultsfromTRFQandSANS,0.31molecules/C.Therefore,correctionto25Cforaggregationnumbersmeasuredat20and21wouldinvolvecorrectionsofonlytwotothreemoleculesthatareignored.Thus,allaggregationnumbersquotedfromtheliteraturearetheoriginaluncorrectedvalues.MethodsandMaterialsSigmamolecularbiologyreagentSDS,99%(lotno.46F-0543),wasusedasreceived.Thesurfacetension,derivedfromthedrop-volumemethod,vsconcentrationplotofthisreagent,giveninFigure1ofref3,showednominimumnearthecmc.DistilledwaterwaspurchasedfromArrowhead.Sodiumchloride99%A.C.S.reagentwaspurchasedfromAldrichandusedasreceived.Asolutionof5DSE(Sigma,asreceived)inmethanol(Spectrum,reagent,A.C.S.,asreceived)wasdistrib-utedtovialsbyweight,driedwithastreamofnitrogengas,capped,andstoredinthefreezeruntilneeded.Afteravialwaswarmedtoroomtemperature,SDSandwaterwereaddedtoprepareamothersolutionofthedesiredconcentrationsofSDSand5DSE.Thissamplewasgentlystirredforseveralhoursbeforeaddingfurtherdistilledwaterand/oraddingsalttothemothersolutiontoproduceaseriesofsamplesinwhich[SDS]or[NaCl]wasvaried,respectively.Thismethodofsamplepreparationensuresaconstantspin-probe/surfactantmolarratiowhether[SDS]or[NaCl]isvaried,thusminimizingtheeffectofvariationsofspinexchangeinteractions(seeAppendix).Afterbeinggentlystirredforafewmoreminutes,thesamples,notdegassed,weredrawnintoglasscapillariesandsealedwithagastorch.Thesamplecapillarywasthenplacedwithinaquartztube,whichwasplacedintheDewarwithinthecavity.ThetemperaturewascontrolledwithaBrukernitrogenflowunittoCandmeasuredwithathermocoupleplacedjustabovethecavityinpositionCofFigure1ofref27.Therationaleforusingthespinprobe5DSEhasbeendiscussedinpreviouswork.Detailsoftheeffectofvaryingthespin-probeconcentrationaregivenintheAppendix.EPRspectraweremostlytakenatX-bandusingaBruker300ESPspectrometer.Alargemodulationamplitudeof1Gwasemployed,takingadvantageofthefactthatonlytheinhomogeneous(Gaussian)componentisbroadened.Atlowmodulationamplitudes,5DSEhasaGaussianlinewidthinSDSmicellesof0.800.02G,independentoftheaggregationnumber.TheGaussianlinewidthaveragedoverthecentrallinesofallthespectrainthisstudywasfoundtobe0.02;thus,themodulationbroadenedtheGaussiancomponentby0.490.03G,inexcellentagreementwitharecentstudyshowingthattheGaussiancomponentisbroad-ened(inquadrature)by/2whereisthepeak-to-peakvalueofthemodulationamplitude.AfewspectraweretakenatL-band(1.28GHz)usingahome-builtbridge,aloop-gapandaVarianmagnet.Thefieldsweepofthemagnet,describedpreviously,wascalibratedwitha0.1mMsolutionofFremy'ssaltin50mMpotassiumcarbonateusingtherecentlyredeterminedhyperfinecouplingconstantof13.056Gat25Figure1showsEPRspectraof5DSEin500mMSDSatCmeasuredatX-band(9.42Ghz,Figure1a)andL-band(1.28GHz,Figure1b).TheresonancefieldsofthelineswereestablishedbyfittingthespectratoaGaussianLorentziansumfunctionasdescribedpreviously.Weshowedpreviouslyforlow-noisespectra,thisprocedurereproducesthepositionsatwhichthefirstderivativelinescrossedthebaselinetobetterthan1mG,providedlinearinterpolationbetweendatapointsisused.ABrukerNMRGaussmeterwasusedinthe1mGresolutionrangetomeasurethesweepwidthofeachspectrum.Inanexperimentspanninganhourortwo,the50Gsweepwidthwasreproducedtoabout34mG.Ameanvalueofthesweepwidthinanyonerunwasusedtocompute,whichistakentobeone-halfthedifferenceintheresonancefieldsofthehigh-andlow-fieldlines,ingauss.Notethatvaluesofarereproducedtowithinabout12mGdespitethefactthattheintervalbetweenconsecutivepointsisontheorderof12mG.ForthespectruminFigure1a,atX-band,0.002G,andinFigure1b,atL-band,0.008G,wheretheerroratX-bandrepresentsstandarddeviationsfromfivespectrausingthesamesample.TheuncertaintyatL-bandisdominatedbycorrectionsformagneticfield-sweepnonlin-Wedefinethespacingsbetweenhyperfinelinesas,wheredenotestheresonancefieldsofthethreelines:1,0,andcorrespondingtothelow-,center-,andhigh-fieldlines,respec-tively(seeFigure1).Measurementsofthesecond-ordershiftparameteryield0.001GforthespectruminFigure1a,whichistobecomparedwiththestaticMicelleSurfaceHydrationModelJ.Phys.Chem.B,Vol.102,No.50,1998 shiftof0.068G;thus,adeparturefromthestaticshiftisobserved.Thistypeofasymmetryiscommonlyobservedinnitroxidespectraastherotationalmobilityslows.Thisasymmetryisdominatedbyashiftofthehigh-fieldline(1)towardthecenterastherotationalcorrelationtimeincreasesinresponsetodecreasingtempera-Denotetheshiftinwithrotationalcorrelationtimebyrespectively.Inan85%OsolutionexperimentalvaluesofGand0.41Gwerefoundasthetemperaturewasdecreasedfrom26.1to11.2Coverwhichrangefrom0.32to1.1ns.Theoreticalcalculationsrangedfrom0to0.007and0.22to0.46Goverthissametemperaturerangedependinguponmodelemployed.Thus,experimentallyandtheoretically,issmall,aboutanorderofmagnitudesmallerthan,andontheorderoftheexperimentaluncertainties.WethereforeneglecttheshiftToestimatetheuncertaintyinneglecting,wenotethattherotationalcorrelationtimereachesamaximumof0.84nswhenSDSisvariedovertherange[SDS]200mM.UsingappendixCofref32for0.84ns,wecalculate4mG.Thisuncertaintywouldintroducearelativeuncertaintyinthevalueof)calculatedfromeq8belowoflessthan0.003,negligiblecomparedwithothersourcesofOurexperimentsareatconstanttemperature;however,asthemicellegrows,therotationalcorrelationtimeincreases;themagnitudeofincreases,whichmeansthattheopera-tionaldefinitionofdecreasesfasterthanthetruehyperfinecouplingconstant.Therefore,partofthedecreaseinisduetodecreasedpolarityandpartduetothedynamicshiftWehavetworelatedbutdifferentobjectivesinthiswork.Thefirstistoprovidearapid,precisemethodofdeterminingtherelativeaggregationnumbersofSDSmicelles.Thesecondistodemonstratethatasimplemodelgivesasatisfactoryaccountoftheobservations.Toachievethefirst,wehavechosentoreportthevariationofisthequantitymostfamiliartoexperimentalistsandvarieslinearly.Thesameconclusionswouldbereachedwereweto;however,doesnotvarylinearlywithmakingthecalculationsmorecomplicated.Toachievethesecondobjective,wecapitalizeonthefactthatissmallandusetocomparetheexperimentalpolaritywithamodel.Second-orderdynamicshiftsfornitroxidesarepredictedtobeweaklydependentuponmicrowavefrequencyintherangeofrotationalcorrelationtimespertinentto5DSEinSDS.Thisseemstobethecase,sincethemeasurementsinFigure1,takenat9.42and1.28GHz,yieldanegligiblediscrepancyinTheEPRspectrumof5DSEinSDSmicellesisanexcellentVoigtshapewithinhomogeneousbroadening(Gaussianlinewidth)equaltothatinhomogeneoussolvents.Thismeansthattheenvironmentreportedbythisspinprobeisaneffectiveenvironmentaveragedbytherapid,almostisotropicmotionoftheNOmoiety.Astaticdistributionofthespinprobeindifferentregionsofthemicellewouldleadtofurtherinhomo-geneousbroadening,whichwouldbeeasilydetectedandisnotDependenceofFigure2showstheresultsofthreetypicalexperimentsatX-bandinwhichtheconcentrationofSDSwasvariedfromjustabovethecmc8.3mMtonear500mMintheabsenceofaddedNaCl.AlsoincludedinFigure2isthedatum(filledcircle)fromthespectruminFigure Figure1.EPRspectraof5DSEin500mMSDSmicelleswithoutaddedsaltat(a)X-bandand(b)L-band.Thedefinitionsofthehyperfine,andareshown.ThevaluesofatX-andL-bandarethesamewithinexperimentalerror. Figure2.asafunctionofaggregationnumberbyvaryingtheSDSconcentrationforthreetypicalrunsatX-band(opensymbols)andthemeasurementderivedfromFigure1batL-band(filledcircle).ThesedatacoveraverywiderangeofSDSconcentrationsfromjustabovethecmc8.3to500mM.Thereproducibilityisgenerallylessthanthesizeofthesymbols;however,thecorrectionforspin-probeconcentration(Appendix)isabouttwicethesizeofthesymbolsnearthecmc.Thestraightlineistheaverageoflinearleast-squaresfitstoallexperiments,i.e.,byvaryingeithertheSDSortheNaClconcentra-tions.Thedashedlineisthesamelinearfit,however,plottedagainsttheaggregationnumbercalculatedfromeq5byneglectingthetermtermfree,showingaseveredeparturefromlinearityatlowaggregationnumbers.Theaggregationatthecmc49.5,isindicated.J.Phys.Chem.B,Vol.102,No.50,1998Balesetal. 1bmeasuredatL-band.InFigure2,isplottedvscalculatedfromeq5.Thereproducibilityisgenerallylessthanthesizeofthesymbols;however,thecorrectionforspin-probeconcentration(seeAppendix)isabouttwicethesizeofthesymbolsnearthecmc.Thestraightlineistheaverageoflinearleast-squaresfitstoexperiments,i.e.,thoseinFigure2inwhich[SDS]isvariedandthoseinFigure3andothersinTable2inwhich[NaCl]isvaried.Thedashedlineisthesamelinearfit,however,plottedagainsttheaggregationnumbercalculatedfromeq5byneglectingtheterm[SDS],showingaseveredeparturefromlinearityatlowaggregationnumbers.Theaggregationnumberatthecmc49.5,isindicatedinFigure2.Figure2showsthatcontinuestobelinearwiththesameslopetoveryhighdetergentconcentrations.Table2detailstheeightexperimentscarriedoutinthiswork.Figure3showstheresultsofthreeexperimentsinwhichthesaltconcentrationwasvariedwith[SDS]fixedand,forcomparison,oneinwhichthedetergentconcentrationwasvaried.Clearly,thevariationofisthesamewhethersaltisaddedordetergentincreased.Forexample,thedatumindicatedbythefilledcirclenear99isderivedfrom[SDS]500mM[NaCl]0whiletheopensquarethatisalmostcoincidentwiththefilledcircleisderivedfrom[SDS]mMand[NaCl]118mM.ThelinearportionsofthecurveswerefittostraightlinesasTheresultsaretabulatedinTable2.Withintheexperimentaluncertainties,theslopesofthelinearcurvesarenotaffectedbythecorrectionforspin-probeconcentrationdetailedintheAppendix.Thevaluesof(0)werecorrectedusingeq17oftheAppendix;however,onlythoserequiringlessthan6mGcorrectionareincludedinTable2.ThemeanvaluescomputedfromtheresultsinTable2inverselyweightedbytheirownvarianceare0.009Gand0.02mG/molecule.ThestraightlinesinFigures2and3areplotsofeq6withthesevalues.Thenumericalvaluesof(0)andineq6dependonthevaluesof,whichhave,thusfar,beencalculatedusingeq5employingaconstantvalueof.Thefollowingquestionarises.Towhatextentdoesusingtheempiricalformofeq4,andspecificallyusingtheparticularparametersineq5,affecttheresults?Toexplorethisquestion,weturntosomevaluesofmeasuredbyavarietyoftechniquesoverawiderangeofvaluesof[Na.Weproceedbyfittingourresultstoanempiricalfunctionof[Na,yieldingthefollowing:Plotsofourresultsasafunctionofof+]aq}0.25areidenticalinformtoFigures2and3;however,nownoreferenceismadetoanassumedvalueof.Equation7allowsustointerpolatevaluesofthatwouldbemeasuredundertheconditionsoftheexperimentstakenfromtheliterature.Fivesetsofvaluesweretakenfromtheliterature.Foreachvalueof,eq7wasusedtocalculatethevalueoffromthecorrespondingvalueof[Na.Theseliteraturevaluesofwerethenfittoalinearequationoftheformofeq6.TheresultsofthesefitsarepresentedinTable3.Thecoefficientsofcorrelationshowthatisagoodlinearfunctionofdirectlymeasuredvaluesof.Theunweightedmeansfromthefiveexperimentsyield0.042Gand0.35mG/molecule,wheretheuncertaintiesarethestandarddeviations.Thus,both(0)andarefoundtobethesamewithinexperimentalerrorwhetherderived Figure3.asafunctionofaggregationnumberbyvaryingtheNaClconcentrationfor50mMSDS()and100mMSDS().Forcomparison,onerunvaryingtheSDSconcentrationisincluded(Thestraightlineistheaverageoflinearleast-squaresfitstoallexperiments,i.e.,byvaryingeithertheSDSortheNaClconcentrations.Theratherabruptdeparturefromlinearityatabout130isattributedtothesphererodtransition. TABLE2:SummaryofExperimentsandLinearLeast-SquaresFitstoEq6fortheSpinProbe5DSEinSDSComputedfromEq5( mM[NaCl]mMA0(0)G@A0/@NAmG/moleculerb25-200c0-3.96(0.120.999(4)495015.5080.200.995(6)200015.4910.050.999(3)20000.040.999(10)20000.200.994(7)100031715.4950.030.999(4)10000.030.999(9)5000.060.999(10)Errorscalculatedfromtheleast-squaresfittoeq5.ofcorrelationusingthenumberofexperimentalpointsindicatedintheparentheses.Datafromref9.FieldsweepcalibratedwithNMRgaussmeter.Fremy'ssalt. TABLE3:FitsoftoEq6fortheSpinProbe5DSEinSDSMicellesUsingExperimentalValuesofTakenfromtheLiterature lightscattering4.330.996(5)64.260.991(3)363.960.949(7)373.790.993(5)383.450.976(4)4mean15.488measuredat25.0Coefficientofcorrelationwiththenumberofpointsintheparentheses.At21Correctedbyisopiesticdistillation,25Small-angleneutronscattering,25resolvedfluorescencequenching.Pyreneexcimerquenching,25Pyrenequenchedbymethylviologen,20C,withvaluesofrecalculatedusing[SDS]ratherthancmc +]aq}0.25(7)MicelleSurfaceHydrationModelJ.Phys.Chem.B,Vol.102,No.50,1998 directlyfrommeasuredvaluesoforfromeq5.Theconclusionisthateq6givesthecorrectdependenceofwithanuncertaintydominatedbytheknownvaluesofInTable4,wepresentthefitsofsomeliteraturevaluesoftoeq4holdingconstant.Thefitstoeq4arerathergood,whichshowsthateq4isagoodempiricaldescriptionoftheaggregationnumbersofSDSevenatsurfactantconcentrationsabovethecmc;however,thereisascatterintheabsolutevalueoftheaggregationnumber.Forexample,theunweightedmeanvalueandthestandarddeviationoftheresultsinTable2atthe4.Equation4isalsoagoodempiricaldescriptionoftheresultsofotherexperiments.Notethattheprecisionoftheexperimentstodateisnotsufficienttodecideisconstantornot;therefore,allowingtovaryasanadjustableparameterisnotyetjustifiedInFigure3,athighervaluesof[Na,oneobservesarathersharpdeparturefromlinearityatabout130([Na400mM),indicatingthatthepolaritysensedby5DSEnolongerdecreaseswithaddedsaltorsurfactant.Manyauthorsobservedasuddenincreaseofmicellesizeat[NaCl]450mM(134,eq5)attributedtoasphereThus,weattributethetransitioninFigure3tothesuddenincreaseingrowthofthemicellesandrefertothisasthesphererodtransition.Notethateq5nolongerholdsabovethesphererodtransitionpoint,sothevaluesofareplottednear140150areinrealitymicelleswithmuchlargeraggregationnumbers,thus,thetransitionisevensharperthanitappearsinFigure3.Wedidnotstudyanypossibleagingeffectsovershorttimespans.Typically,asamplewasmeasuredthedayafteritwasfirstprepared.Comparingtheresultsfromasamplewith[SDS]200mMand[NaCl]0after1dayand2monthsshowedidenticalvaluesofInthecaseof5DSE,byuseoftheoperationaldefinitionof,thevariationofislinear,soisaconstant.Preliminaryworkusingotherspinprobesandmeasurementof5DSEatL-bandrevealslightcurvaturesinthevariationsofDependenceofonthePolarityIndexThevalueof(ratherthantominimizedtheeffectsofthesecond-orderdynamicsshifts)wasmeasuredasafunctionofpolarity.Solutionsof5DSE(0.05mM)werepreparedinaseriesofmixturesofmethanolandwater.Foreachmixture,thevalueofthepolarityindexC)wascalculatedfromalinearleast-squaresfitofthedatagiveninTable1ofMukerjeeetal.forwater/methanolmixturesat25Casfollows:,whereistheweightfractionofmethanol.Thislinearfitdeviatesbyamaximumof0.4%fromthecorrectvaluesovertherange0.828,whichcorrespondstoThemeanvaluesoffivemeasurementsofineachmixtureareshowninFigure4versusThestandarddeviationsaresmallerthanthesymbols.Fittingthedatato0.025Gand0.044Gwithcoefficientofcorrelation0.998overtherange0.69.Mostofthemeasurementsleadingtoeq8weremadeatroom-temperaturePreviously,wefoundonlyvariesbyabout0.2C,socorrectionsto25Carenegligible,andafewmeasurementsat25.0Cconfirmedthisfact.DependenceofthePolarityIndexC)ontheAggregationNumberFigure5showstheresultsofcomputingvaluesofthepolarityC)inSDSmicellesfrommeasuredvaluesoffortwooftheexperiments.TheresultsofalloftheexperimentsfallverynearthoseinFigure5withscattersimilartothatinFigures2and3.Itisimportanttonotethatelectricfieldsnearthesurfaceofthemicellearenotexpectedtoinfluencethevalueof,sincerapidrotationofthenitroxidemoietyaveragestheeffectsofthesefields.HydrationModelWeconstructasimplemodelbasedonaclassicalpictureofthemicelleashavingahydrocarboncorewithverylittlewaterDrawingupontheworkofMukerjeeandco-wesupposethatthespinprobeisadsorbedontothemicellesurface.Thespinprobe5DSEisknowntoexecuterapid,almostisotropicmotioninanSDSmicelle;thus,wesupposethatthenitroxidemoietyrandomlyandrapidlysamplesallportionsofthepolarlayersurroundingthehydrocarboncore. TABLE4:FitstoEq4ofExperimentalValuesoffromtheLiterature( eq549.50.250.00838lightScattering51.90.2120.008140.995(5)650.10.2360.00820.996(3)3644.80.2140.00830.987(7)3747.30.2590.00830.995(5)3854.20.2470.00830.978(4)4CoefficientofcorrelationwiththenumberofpointsintheBestfittosodiumoctyl,nonyl,decyl,undecyl,anddodecylsulfateatAt21Correctedbyisopiesticdistillation,25Small-angleneutronscattering,25resolvedfluorescencequenching.Pyreneexcimerquenching,25Pyrenequenchedbymethylviologen,20C,withvaluesofrecalculatedusing[SDS]ratherthancmc Figure4.asafunctionofthepolarityindexC)inaseriesofmethanolwatermixtures.Thestraightlineisalinearleast-squaresfittothedata. C)(8)J.Phys.Chem.B,Vol.102,No.50,1998Balesetal. Weproposethatthephysicalbasisforthedecreasein)withmicellesizeisduetoadehydration(persurfactantmolecule)ofthemicellesurfaceasthemicellegrows.Thisisasimplegeometriceffect:asthemicellegrows,theareaperheadgroupdecreases,leavinglessroomforwatermolecules.Decreasingwatercontentwouldleadtoadecreasingpolaritythatwoulddecrease.Foradiscussionofmicellehydration,seeref41andreferencestherein.WenotethatessentiallythismechanismwasadvancedmanyyearsagobyLianosandZanatoaccountforthesystematicincreaseintherelativeintensitiesofthethirdtofirstvibronicbandsinthefluorescenceemissionspectrumofpyrene.Weassume(1)thatasimpletwo-shellsphericalmodel,ahydrocarboncoreandapolarshell,isadequatetodescribethehydrationofthepolarshell.Additionally,weassume(2)thenitroxidemoietyof5DSEresidesinthepolarlayer,undergoingrapidrotationalmotionthatallowsthenitroxidetosampleallregionsofthelayerandreporttheaveragepolarity.Finally,weassume(3)thatthethicknessofthepolarshellisapproximatelyconstantasthemicellegrows.ThisassumptionhasbeenpreviouslyemployedintheinterpretationofSANSandseemstobereasonable,sincethethicknessofthepolarshell(5…)isapproximatelyequaltothediameteroftheSONaheadgroup.Inthismodel,themicelleiscomposedofaninnersphericalhydrocarboncoreofradiusandanouterconcentricpolarshellofthicknessistheradiusofthemicelle.Thecorecontainsthehydrocarbontailsofthesurfactantmolecules,exceptperhapsforthefirstmethylenegroup,andthepolarshellcontainstheheadgroups(OSO),afractionofthecounterions(Na),andwater.Anellipsoidofrevolutionisoftenemployed,aspherebeingaspecialcase.thegeometryofthemicelleischaracterizedbytheradiusofthehydrocarboncore,,andtheradiusofthemicelle,Thevolumeofthecoreistakentobe,whereisthevolumeoccupiedbythesaturatedhydrocarbonchain,calculatedaccordingtoTanford(p52ofref48)asisinunitsof…isthenumberofcarbonatomsperchainthatareembeddedinthemicellecore.Thus,thecoreradiusisfoundfromThevolumepersurfactantmoleculeinthepolarshell,,isgivenbyThereisdisagreementintheliteratureabouthowmanymethylenegroupsareªwetº,i.e.,whatthevalueofoughttobe.Wetake12,inkeepingwithadoptingthesimplestapproach,yielding350…,whichcompareswellwithCabane'sestimateof346…TheSANSresultsofCabaneshowedthatthethicknessofthepolarshellis5….WedenotetheaveragenumberofwatermoleculesresidingO),whichwerefertoasthehydrationnumberpersurfactantmolecule.Takingthevolumeperwatermoleculetobe30…thevolumefractionofoccupiedbywaterisgivenbyisinunitsof…TherehavebeenanumberofinvestigationsofthehydrationofSDSmicelles,soideally,wewouldfixthevalueO)fromothermeasurementsandproceedwithnoadjustableparameters.Unfortunately,theestimatesofvarybetween8and12whenderivedfromhydrodynamicandmuchlargervalueshavebeensuggested.ThepreviousmodelsdonottakeintoaccountthefactthatO)couldvaryasthemicellesgrow,soanaveragevalueO)wouldemerge.OurmodelpredictsadecreaseinthevalueofO)asthemicellegrows,sooneneedstoataparticularalueofNandthencomputeO)asafunctionofLetusillustratebyusingasareferencethe69mMSDSsystememployedbyCabane,whosuggestedAccordingtoeq5,63,andevaluatingeqs10and11gives397….Ifwedivideintothevolumeoccupiedbywaterandavolumeinaccessibletowater,,then397…157…Thesamecalculationsfor9at63gives127…,andfor9.6,theyleadto109…Forcomparison,thevolumeoccupiedbythesodiumsulfateheadgroupisabout66.4…somewhatsmallerthantheseestimatesofO).Insistingonavalueof66.4…leadsto11.0at63.Itisimportanttonotethatonemayspecifyeitherorthevalueofforagivenvalueoftofixthemodel.Thevalueofthepolarityparameter,combiningeqs12and13,isgivenbyToavoidanadjustableparameter,wemakethereasonableassumptionthatasthemicellegrows,theinaccessiblevolume Figure5.ExperimentvaluesofthepolarityindexC)oftheSDSmicellesurfaceasmeasuredbythespin-probe5DSEversustheaggregationnumber.Onlytwooftheexperimentsareshownforclarity;theothersoverlaythesedatawithscattersimilartothatinFigures2and3.Thesolidlinesarecalculatedfromthegeometricalhydrationmodeldescribedinthetext.Thenumbersnexttothesolidlinesindicatethenumberofwatermoleculespersurfactantmoleculeatthereferencesizeof 3Rc3(10)Vp)4ð )(11))/30(13)MicelleSurfaceHydrationModelJ.Phys.Chem.B,Vol.102,No.50,1998 isconstant.Therefore,thepolarityindexispredictedtodecreaseaccordingtoeqs10,11,and14.TheseresultsareplottedassolidlinesinFigure5togetherwiththeexperimentalresultsoftwoofthetypicalexperiments.ThenumbersneareachsolidlinedenotethevalueofO)at63forthatline.Clearly,thetheoreticalcurvefor9atisinexcellentagreementwiththeexperimentalresults.Thehydrationnumbermaybecalculatedbycombiningeqs10,11,and13.ThesehydrationnumbersvarywithmicellesizeasplottedinFigure6forthecaseof127….Thetotalwaterassociatedwiththemicellemaybecalculatedbymultiplyingthehydrationnumberpersurfactantmoleculebytheaggregationnumber;i.e.,totalwaterO).ThetotalwaterversusmicellesizeisalsoplottedinFigure6usingtheright-handordinate.ThefilledrectangleinFigure6representsthehydrationnumbersestimatedbyTokiwaandfrommeasurementsofsedimentation,diffusion,viscosity,anddensityofSDSintheconcentrationrangefromcmcto87mMin100mMNaCl.OfthevariousinvestigationsofSDS,thedataoftheseauthorsseemtobethemostappropriateforthefollowingreasons.First,theirchoiceof100mMNaClisanexcellentcompromisebetweentheneedtoreducetheelectroviscouseffectandtheneedtohavesmallenoughmicellesinordertoassumeasphericalshape.Onemustseparatetheeffectsofshapeandhydrationinordertoderiveahydrationnumberfromhydrodynamicmeasurements.Second,holding[NaCl]constantat100mMandvarying[SDS]fromthecmcto87mMmeansthattheaggregationnumbervariesonlybyabout6%aboutthecentralvalueof98.Thisminimizestheproblemofnottakingintoaccountthevariationofwithmicellesize.ThehorizontalextentoftherectangleinFigure6representstherangeofintheirmeasurements.Third,TokiwaandOhkiestimatedthehydrationnumbersintwoways:(a)fromsedimentationanddiffusiondataand(b)fromreducedviscositydata,findinggoodagreementbetweenthetwomethods.ThetopandbottomoftherectangleinFigure6correspondtothesetwodifferentmethodstoestimatethehydrationnumber.WeemphasizethatthenumericalvaluesofC)andO)predictedbyeqs914aredependentonthemodelemployedforthemicelleaswellasthenumericalvaluesoftheparameters,notablyandthethicknessofthepolarshell.Therefore,Figure5oughtnottobetakentoprovethatahydrationnumberat63of9isbetterthan,8.Theimportantpointisthat,afterfixingoneparameter,asimplemodelwithreasonableassumptionsgivestherateofdecreaseofpolaritywithmicellesizeconsistentwithexperimentalresults.RelativeAggregationNumbers.Themicropolaritysensedbythe5DSEprobedependsdirectlyontheaggregationnumberoftheSDSmicelleandnotontheionicstrengthofthesolution.Thisisdemonstratedbythefactthatforagivenvalueofattainedusingdifferentcombinationsof[NaCl]and[SDS],themethodyieldsthesamevalueof.Thismeansthatthepolarityofthesurfaceofthemicelleisindependentof[NaCl]exceptthroughthesalt'sroleindetermining.Themicropolarityforagivenvalueofisalsoindependentofthenumberofmicellesinsolutionwithinthelimitsoftheseexperiments;i.e.,interactionsbetweenmicellesdonotaffectthesurfacepolarity.Thelimitsareconsiderable.Consider,forexample,thetwopointsinFigure3near99,correspondingto[SDS]500mMand[NaCl]0versus[SDS]50mMand[NaCl]118mM,respectively.Thesetwopointscorrespondtoto)5.0and0.5mM,respectively.AtthehigherSDSconcentration,themicellesaregettingtobequitecrowded.Byuseofasimplecubiclattice,thecenter-to-centermicelledistanceis69…,comparedwithtwicethemicelleradius250.4….Forachangeinaggregationnumber,,eq6yieldsachangeinhyperfinecouplingconstant,For5DSE,0.02mG/molecule,andsincethespacingbetweenhyperfinelinesof5DSEcanbemeasuredwithaprecisionof13mG,thismeansthatrelativevaluesofmaybedeterminedfromeq15toaboutonemoleculepermicelle.Theprecisionwithwhichabsolutevaluesofbedeterminedislimitedinalmostallcasestotheaccuracytowhichtheseaggregationnumbersareknownandnotbytheexperimentaluncertaintiesinthemeasurementof.TheonlyexceptionariseswhenoneattemptstomeasureinsolutionsverynearthecmcwheretheuncertaintiesdiscussedintheAppendixcouldenter.WeachievehighprecisioninthemeasurementsofeveninrathernoisyspectrabyfittingthespectratoaVoigtlineshape.Withoutfitting,itisnotdifficulttoachieveprecisionsontheorderof0.01Gonspectrashowinggoodsignal-to-noiseratios,butthiscanincreaseto0.03Gonnoisyspectra.Withtheselatterspectra,theuncertaintyinrelativevaluesofreachesabouteightmolecules,stillsufficientforsomepurposes.RelyingonaHallprobecalibratedsweeprangeisprobablyadequateforrelativevaluesof(relativevaluesof);however,absolutevaluescanbeinseriouserror.Forexample,withourunit,a50Gsweeprangecorrespondstoabout50.2G,whichgivesanerrorofabout0.06Gin,i.e.,anuncertaintyintheabsolutevalueof15molecules. Figure6.Numberofwatermoleculespersurfactantmolecule(left-handordinate)andthetotalwaterassociatedwiththemicelle(right-handordinate)asafunctionoftheaggregationnumber.Thecurvescorrespondtofixing127…,i.e.,arederivedfromthecentercurveinFigure5.TherectanglerepresentsestimatesoffromthehydrodynamicdataofTokiwaandOhki.Thehorizontalextentoftherectanglecorrespondstoarangeof,andtheverticalextentspansthedifferenceintwomethodstoestimateMukerjee'stheoreticalestimatebasedononlyprimarysolvationoftheionsis5,nearthelowerlimitofthepresentobservationsatthesphererodtransition. J.Phys.Chem.B,Vol.102,No.50,1998Balesetal. Thephysicochemicalpropertiesofmicelleschangerathersubtlywithsize,makingthemdifficulttostudybymanymoredirecttechniques.Althoughthepresentapproachsuffersfrombeingindirect,itdoesofferasimple,precisemethodtolookforsmallchangesinInparticular,thepotentialtostudysubtleeffectsof(smallamounts)ofvariousadditivesisinteresting.TheinfluenceofanumberofadditivesontheaggregationnumbersofSDSmicelleshasalreadybeenstudiedbyAlmgrenandco-work-usingtime-resolvedfluorescencequenching(TRFQ).Particularlyinterestingwillbelong-chainalcoholswherepresumablytheOHgroupofthealcoholwouldenterintothecalculationofthepolarity.Itwillbeveryinterestingtoseeifthismethodiscapableofdetectingthesmallchangesinwithtemperature.Asapracticalmatter,theinterferenceofspinexchangeasdescribedintheAppendixwouldneedtobetreatedwithcarebecausethespinexchangefrequencyisstronglytemperature-depend-FurtherEvidenceThattheMicropolarityofSDSMicellesDecreaseswithIncreasingSize.Thereisampleliteraturethatexploitsthesolvent-dependentpropertiesofvariousprobesinordertostudythepolarityofSDSmicelles(see,forexample,ref56,especiallyS.Table2oftheSupportingInformation).Themodeladvancedherepredictsthatanymoleculepossessingsolvatochromicpropertiesresidingforasignificantfractionofthetimenearthemicellesurfaceoughttoshowadecreaseinpolarityasthemicellesgrow.Toourknowledge,theonlysuchpropertythathasbeensystematicallystudiedasafunctionofmicellesizeistheso-calledI/IIIratiointhefluorescenceemissionspectrumofpyrene.Figure7showsdatatakenfromtheliteratureplottedasafunctionoftheaggregationnumber.InFigure7,thedataofLianosandZana,bycircles,wereobtainedbyvaryingtheconcentrationofNaClina70mMSDSsolutionandthevaluesofarethosemeasuredbythesameauthorsemployingTRFQmeasurementsofthepyreneexcimer.ThedataofSiemiarczukandWare,representedbysquares,wereobtainedbyvaryingtheSDSconcentrationintheabsenceofsaltwherethevaluesofcomputedfromeq5.TheI/IIIratioisknowntogenerallydecreasewithdecreasingpolarity,sothedatainFigure7maybeinterpretedasadecreaseinpolarityasthemicellesgrow.Unfortunately,thevaluesoftheratioI/IIIarenotwellreproducedbydifferentlabs,aproblemthathasbeenattributedtoinstrumentaldifferences.Forexample,measurementsoftheratioforpyreneinwaterfromvariouslabsleadtoastandarddeviationofabout9%,whilereproducibilityisabout1Inviewofthis,thedataofSiemiarczukandWarehavebeenscaledupwardby8%sothatthesalt-freedatumofLianosandliesontheinterpolationbetweentwosalt-freedatapointsofSiemiarczukandWare.Thisplacesthetwodatasetsonthesamefooting;however,theabsolutevalueoftheordinateisuncertainbyabout8%.DerivinganestimateofthepolarityfromtheI/IIIratioinvolvesfurtherdifficulties,sincethereisalackofunderstandingabouttheprobesolventinteractions.PlotsoftheI/IIIratiovsthepolarityparameterC),forexample,showpoorcorrelation.Thus,withthepresentprecisioninthemeasurementsoftheI/IIIratioandthelackofasoundtheorywithwhichtoproceedtoanonempiricalpolarityscale,notmuchmorecanbedonewiththedatainFigure7;nevertheless,thesimilarityinFigures7and3iscompellingindeed.Theconclusionisthatthepolaritydecreaseswithmicellesizewhenmeasuredwithaverydifferentmolecule,pyreneratherthan5DSE,usingadifferentpolarityparameter,andthattherateofdecreaseissimilarwhetherNaClorSDSisvaried.Inpracticalterms,theI/IIIratiowouldnotbeaverysuitableavenuetostudyrelativemicellesizes.Inprinciple,theopticalspectraofthenitroxideprobescouldbeusedtopursuethesamekindofstudiespresentedhere;however,inpractice,ratherlargeconcentrationsoftheprobearerequired.Therelativeprecisionoftheopticalmeasurementsisabout0.3nm/51nm0.006comparedwith0.001forEPRmeasurementsof.EPRenjoysthefurtheradvantageofnotrequiringsampletransparencythatmighthinderopticalmeasurements.HydrationoftheSDSMicelleSurface.Thehydrationmodelpredicts,strictlyonageometricbasis,adryingofthemicellesurface(persurfactantmolecule)asthemicellegrowsbecauselesswaterwillfitwithinthepolarshell.CreditforthisbasicideaoughttogotoLianosandZana.WeconsiderthemodeltobeasuccessbecauseitpredictstheslopeofthepolarityvsaggregationnumberingoodagreementwithexperimentalresultsinFigure5.TheratherremarkablygoodfitbetweentheoryandexperimentinFigure5isperhapsfortuitousgiventhesimplicityofthemodel.Furthermore,thequantitativetransformationfrommeasuredvaluesofvaluesofthepolarityindex,C)involvesanuncertainty(discussedbelow)oftheeffectofions.ThisuncertaintycouldshifttheexperimentalpointsinFigure1upordown;however,theslopeisprobablynotaffectedverymuch.Thus,atthisstage,onecannotrelyontheabsolutevaluesofO);however,itdoesseemclearthatadecreaseinvaluesofO)asthemicellegrowsisaviableworkinghypothesis.ThepredictedbehaviorofC)vsisquitesensitivetotheassumedvalueofthehydrationnumberpersurfactantmolecule,ascanbeseenbycomparingthecurvesinFigure5thatdifferbyonlyonewatermoleculeorless.Themodeloughttoberathereasilyextendedtomicellesformedfromsurfactantsofanychargestate,providedthatasphericalshellmodelisapproximatelycorrect.Ifthisprovestobetrue,thenmeaurementsofwillleadtopreciserelativeaggregationnumberdeterminationsformanymicelles.In Figure7.Intensityratiosofthefirstoverthethirdpeakinthefluorescencespectrumofpyreneasafunctionoftheaggregationnumber.DatafromtheliteratureareobtainedbyvaryingtheconcentrationofNaClina70mMSDSsolution)orbyvaryingtheSDSconcentrationintheabsenceofsalt).Thedataarescaledtoaccountforinstrumentaldifferences.Seetext.MicelleSurfaceHydrationModelJ.Phys.Chem.B,Vol.102,No.50,1998 practicalterms,themethodprovidesameanstomonitorrelativemicellesizesasexperimentalconditionsarevaried.ThemodelisconsistentwiththelevelingofthepolarityatthesphererodtransitionpointevidentinFigure5near130.Althoughthemicellecontinuestogrow,infact,muchfasterthantheabscissaindicatesabove130,sinceeq5nolongerappliesabovethetransition,thevolumeperheadgroupisnotexpectedtovarymuchifthegeneralpictureofanelongatingrodiscorrect.ThetotalwaterassociatedwiththemicelleincreasesasshowninFigure6(right-handordinate).Thetotalwatervariesonly15%aboutitsmidpointvalue,soalthoughthedryingpersurfactantmoleculeisrathersevere,theoverallhydrationchangeismoremodest.ThismeansthatanymethodtodetermineO)versusbasedonameasurementofthetotalwaterassociatedwiththemicellewouldneedtoberatherprecise.Wehavepresentedthehydrationofthemicellesurfaceasageometricphenomenonwiththecounterionconcentrationenter-ingindirectlyastheagentthatdeterminesthemicellesize(eq1).AdecreaseinvaluesofO)asthemicellegrowsalsoseemsphysicallyreasonablefromanotherpointofviewifweviewthemicelleastwoªphasesºanddrawupontheanalogywithion-exchangeresins.Theseresinsswellwithwateraroundthechargedionogeniccentersinanattempttoequalizetheconcentration(activity)oftheionsintheresinandintheaqueousphase(see,forexample,chapter5ofref59).Theswellingisdrivenbytheosmoticpressuredifferencebetweenthetwoªphasesºandisopposedbythemechanicalpressurefromtheinterchaincross-links.Theinterplaybetweentheswellingpressureandthemechanicalpressuredeterminestheequilibriumwatercontent.Iftheionconcentrationisincreasedbyaddingsalt,theionconcentrationdifferenceinthetwophasesdecreases,reducingtheswellingpressureandleadingtoalowerwaterconcentration.Inionicmicelles,thehydrophobiceffectprovidestheforcethatlimitstheªswellingº.Forincreasingvaluesof[Na,furnishedeitherbyhigherdetergentconcen-trationsorbyaddedsalt,thediminishedconcentrationdifferencewoulddecreasetheosmoticpressure,leadingtoadehydration(persurfactantmolecule)ofthemicellesurface.TheagreementbetweenthevaluesofO)derivedfromspin-probemeasurementsandthetheoreticalmodelandthosefromtransportdata,representedbytherectangleinFigure6,supportstheideathatthenumberofwatermoleculesthatmoveasakineticunitwiththeSDSmicelleisthesameasthenumberofwatermoleculesthatarehousedinthepolarshellandenterintoourmodel.ThissupportsthecontentionofMukerjeesuggestedthatadditionalhydrationlayerswerenotimportant.OurmodelisdifferentindetailthanthatenvisionedbyinthatO)varieswithmicellesize.Muker-theoreticalestimateof5,usingprimaryhydrationnumbersof4and1forNaandthesulfateheadgroup,respectively,issmallerthanourvalues(Figure6)exceptnearthesphererodtransition.TotheextentthatourvaluesofO)inFigure6holdupunderfurtherinvestigationandthattakingMukerjee'sestimateof5tobereasonablemeansthatsomeofthewaterinthepolarshellisªfreeºovermostoftherangeof.Wedonotyetknowifitissignificant,butthesphererodtransitionoccursataboutthepointthatªfreeºwaterdisappearsinourmodel,sincetheprimaryhydrationof5isnearourestimateof5.5atthisInthemodel,itwassupposedthatthenitroxidemoietysampledallofthepolarshellandonlythepolarshell.ThefactthatthemodelgivesreasonableresultsforO)supportsthisview.Thespinprobe5DSEisquitehydrophobicexceptfortheNOgroup.ItseemsreasonablethatthehydrophilicNOgroupof5DSEresidesinthepolarshellbecause,inorderforthegrouptorangeveryfaroutsidethepolarshell,eitherthehydrophobicmoietiesattachedtotheNOgroupwouldhavetobedraggedintothehydrophilicregionorthehydrophilicNOgroupwouldhavetoresideinthehydrophobicregion,contrarytothefindingsofMukerjeeandco-workers.Nevertheless,anyrapidmotioninvolvingexcursionsoftheNOgroupintothehydrocarboncoreandintotheaqueousregionwouldinvolveaveragingvaluesofC)nearunityandzero,respectively,andwouldbedifficulttodetect.Forexample,excursionsofequalprobablityintotheseregionswouldyield0.5,whichwouldhavetobeaveraged,withtheappropriatestatisticalweight,withthevaluesofC)appropriatetothepolarshell.ItisclearfromFigure5thatsuchexcursionswouldaffectthereportedpolarityminimally,since0.5isnearthevaluesreported.TheremarksinthepreviousparagraphwouldapplytoanyspinprobecomposedonlyofhydrophobicmoietiesexceptfortheNOgroup.Inparticular,theentireseriesofdoxylstearicacidesterswiththedoxylattachedatvariouscarbonatomsoughttohaveaboutthesamepolarityinanSDSmicelle.Incontrast,thedoxylstearicacids,particularlythoselabelednearthechargedcarboxylgroup,wouldpossessanadditionalhydrophilicmoietythatwouldbeexpectedtoaltertheaveragelocationoftheNOgroup,possiblyleadingtodifferentresults.TestingoftheModelThemodelpredictsthatonlythevolumefractionoccupiedbywaterinthepolarshellisimportant.Withinthecaveatofthefollowingparagraph,thismeansthatthepresenceofcounterionshasonlythesecondaryeffectofoccupyingvolumeinthepolarshellofthemicelle.Thus,exchangingLiforNainSDSispredictedtohaveonlyaminoreffectifmicellesofthesamesizearecompared.Itisimportanttonotethatindevelopingthemodel,wehavenotdistinguishedbetweenªfreeºwaterandwatersofsolvationassociatedwiththecations(seep104ofref59).WehavemadethetacitassumptionthatthepolarityindexC)doesnotdistinguishbetweenthetwotypesofwater;therefore,atthislevelofsophistication,itdoesnotmatterthatahydratedLiionisprobablylargerthanahydratedNaion,sinceweareaccountingforthetotalwaterinthepolarshell.Thus,wepredictthatthevalueofforagivenvalueofwouldnotdependonthecounterionbecause,althoughthetwoionshavedifferentsizes,neitheroccupiesmuchvolume.Note,however,thatforagivenvalueoftheright-handsideofeq5,onefindsasmallervalueofforlithiumdodecylsulfatethanforSDS.Incontrast,ifamuchlargercounterionwereemployed,e.g.,trimethylammonium(140…thespace-fillingaspectmightbedetectable.Obviously,ineverycase,anindependentmeasurementoftheaggregationnumberisneededinordertostudyvstheaggregationnumber.Inaqueousandmethanolicsolutions,dissolvedsaltshavebeenfoundtoproducesmallchangesinthevaluesofsmall,neutral,watersolublespinprobes.WehaveneglectedsucheffectsandindiscussingtheanticipatedresultsusingLiinplaceofNa,wehaveassumedthatsucheffectswouldnotdependonthereplacementofoneoftheseionsbytheother.Anotheraspectofthelithiumversussodiumcounterionexperimentscanbeeasilytested.Thesphererodtransitionthoughttooccurinsodiumdodecylsulfatedoesnotoccurinlithiumdodecylsulfateevenuptosaltconcentrationsof0.81M.Accordingtothemodel,onedoesnotexpectalevelingofthevalueof)athighLiJ.Phys.Chem.B,Vol.102,No.50,1998Balesetal. Aseveretestofthemodelwouldcomefromstudiesoftherestofthenormalsodiumalkylsulfateswithcarbonchainlengthsfrom8to11.Themodelpredictsthatwouldbethesameunderconditionsinwhichthevolumeinthepolarshellpersurfactantmoleculewerethesame,providedtheinaccessiblevolumewerethesame.Evenwiththeuncertaintiessurroundingthesimplemodel,itisreasonabletoassumethatthedetailsofthepolarregionwouldbesimilar,sincetheyhousethesameheadgroup.Therefore,aplotofispredictedtobeauniversalcurvewithnoadjustable,noteven.Eventhequestionofwhetheroughttobethetotalnumberofcarbonatomsorlessthanthetotalwouldnotaffecttheresultsverymuchaslongasconsistencyismaintainedthroughouttheseries.couldbevariedwithvariouscombinationsofsurfactantandsaltcon-centration;thus,theuniversalcurvecouldbetestedoverquitewiderangesofThepolarityofSDSmicellessensedbythespinprobe5DSEdependsonthemicelleaggregationnumberonly,notonthecombinationofsurfactantandsaltconcentrations.One-halfthedifferenceinthehigh-andlow-fieldresonancefields,,varieslinearlywithaccordingtoeq6;thus,theabsolutevalueofmaybedeterminedwithaprecisionlimitedonlybyitsknownvalue,whilerelativevaluesofmaybedeterminedtowithinaboutonemolecule.Theseconclusionsdonotdependontheuseoftheempiricalequationeq5.Amodeldescribingthehydrationofthemicellesurfaceasafunctionofmicellesizeisadvancedthatisinexcellentagreementwithexperimentalobservationsemployingonlyoneadjustableparameter,thehydrationnumberpersurfactantmoleculeatagivenvalueof.Oncethisparameterisfixed,thehydrationnumberpersurfactantmoleculedecreaseswith.Themodelmaybesubjectedtoseverefutureexperimentaltests.ThisworkwassupportedbygrantsfromtheFundacaÄodeAmparoAPesquisadoEstadodeSaÄoPaulo(FAPESPProject94/1111-8),theNIH/MBRSS06GM48680-03,andtheCSUNResearchandGrantsCommittee(toBBandM.P.).A.V.wassupportedasaMBRSUndergraduateResearchAppendix.DependenceofuponSpin-ProbeSpinexchangebetweenspin-probemoleculesleadstothewell-knowneffectofshiftingtheouterlinestowardthewhichwouldappearasanapparentdecreaseinaccordingtoouroperationaldefinition.Inamicellarsolution,toobtaincorrectspinexchangefrequenciesfromtheobservedshifts,properaccountforthedistributionofthespinprobeamongthemicellesmustbetaken(seeeqs1215ofref62andreferencesthereinfordiscussionsoftheproblem).Briefly,micellescontainingonespinprobeyieldunshiftedspectra,thosecontainingtwoyieldashift,thosecontainingthreeyieldalargershift,etc.Therearespinstatisticsaswellasmicelledistributionstatisticsinvolved,andsubtlematterssuchasthefactthatthespinexchangefrequencyisexpectedtobesmallerinalargermicelle.Forthepurposeathand,thiskindofdetailisunnecessary,sincewewishtocorrectfortheshiftratherthandeducethespinexchangefrequencies.Wemakethestandardthatthespinexchangefrequencyisproportionaltothemolarratioof5DSEtomicellizedSDS,.Assumingthatallofthe5DSEisassociatedwiththemicelles,micelles,(-[SDS]free).Usually,experimentsarecarriedoutwithconstantmolarratiosofspinprobetosurfactant,[5DSE]/[SDS],whichisrelatetoasfollows:Formostapplications,itisrathereasytodesignexperimentswithquitesmallvaluesof.ByselectionofmodulationamplitudesontheorderoftheEPRlinewidthandmicrowavepowersthatleadtoslightpowersaturationandbyuseofleast-squaresfittingtechniques,accurateresultsmaybeobtainedfromsamplesofconcentrationaslowas10M.Thismeans,forexample,thatforanexperimentwith[SDS]nearthecmcof8.3mM,avalueofaslowasmaybequicklyattainedwithnoneedforsignalaveraging.Figure8showsfora50mMSDSsolutionwithnoadditionalsalt.Thesolidlineisalinearleast-squaresfitThesecondtermistheshiftduetospinexchange.Combiningeqs17and6showsthatanerrorofaboutonemoleculeresults0.002.Toapplythecorrection,iscalculatedfromtheexperimentalpreparation,[SDS]fromeq5ofref8,andfromeq16and2.13isaddedtotheobservedvalueof.Thiscorrectionmaybecalculatedforeachsampleinaseries;however,thecorrectionoftenwillbethesameforeachmemberoftheseries.Thisoccursforexperimentsthatemployaconstantvalueof,and[SDS]/[SDS]eitherisnegligibleordoesnotvarysignificantly.Ineithercase,isap-proximatelyconstant.Forthesecases,evenifthecorrectionineq17isnotnegligible,onlythevalueof(0)isaffected,notthevalueofReferencesandNotes(1)Croonen,Y.;GeladeÂ,E.;VanderZegel,M.;VanderAuweraer,M.;Vandendriessche,H.;DeSchryver,F.C.;Almgren,M.J.Phys.Chem.,1426.(2)Bezzobotnov,V.Y.;BorbeÂly,S.;Cser,L.;FaragoÂ,B.;Gladkih,I.A.;Ostanevich,Y.M.J.Phys.Chem.,5738.(3)Bales,B.L.;Almgren,M.J.Phys.Chem.,15153.(4)Gehlen,M.H.;DeSchryver,F.C.J.Phys.Chem.,11242.(5)Almgren,M.;Swarup,S.J.Phys.Chem.,4212.(6)Huisman,H.F.Proc.K.Ned.Akad.Wet.,367.(7)Sasaki,T.;Hattori,M.;Sasaki,J.;Nukina,K.Bull.Chem.Soc.,1397. 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